The Term Structure of Interest Rates: The Theories and Their Limits

The term structure of interest rates is the set of SPOT rates one credit quality carries across different lengths of time, read at a single moment. On the invented curve used here those SPOT rates begin at 5.90 per cent a year for one year and reach 7.60 per cent at thirty, on annual compounding. Four readings account for that rising shape, and a single curve cannot separate them.

Underneath that sits a plainer idea. A rate is a price for renting money over a stretch of time, and there is no law anywhere saying the price for one year and the price for thirty years must be the same number. The moment they are allowed to differ, the whole set of them stops being a list and becomes an object with a shape. A shape invites explanation, and four explanations compete for this one.

What is the term structure of interest rates a structure of?

A reader who has not been told exactly what is being explained will hear four explanations of four different things, so the object has to be fixed before any reading of it is offered. Every rate written in this guide is a SPOT rate. A SPOT rate is the rate for money placed today and returned in one lump at one stated future date, with nothing paid in between. Each SPOT rate belongs to one length of time and to no other. The five year SPOT rate says nothing about the third year on its own; it prices the whole five year stretch as a single trip.

Three further conditions are attached and all three matter. Every SPOT rate here belongs to one credit quality, meaning one borrower of unchanging standing across every length of time. Every one of them is read at one moment, so the whole set is a photograph rather than a history. And every one of them is struck on annual compoundingInterest added once a year. An amount held for three years is multiplied by one plus the three year rate three separate times, rather than by a half rate six times., meaning one interest period a year. Change any one of those three conditions and the numbers below stop reproducing.

Now the object itself, and it is smaller than most readers expect. The invented curve running underneath this whole sequence fixes a SPOT rate at six lengths of time and at no others: a one year SPOT rate of 5.90 per cent a year, a two year SPOT rate of 6.25 per cent, a three year SPOT rate of 6.55 per cent, a five year SPOT rate of 6.90 per cent, a ten year SPOT rate of 7.35 per cent and a thirty year SPOT rate of 7.60 per cent. Every one of those six was chosen for teaching, none is taken from any market, and there is no seventh.

The next thing a reader wants is a seventh, so say the last part out loud. There is no four year SPOT rate in this record. There is no nine year one and no twenty nine year one. Each length of time where a rate actually exists is a nodeA length of time at which a rate is actually recorded. Between two nodes this record holds nothing, and nothing is read off the space between them., this record carries six nodes, and the stretches between two nodes are empty rather than smooth. Filling one means choosing a method, and a straight reading and a curved reading disagree, so two accounts working honestly from one record would print two different numbers for the same object. No line is drawn between the points.

Here is the everyday version, and it survives translation better than most. A bank branch displays a board of deposit rates. There is a rate for a year, a rate for two, a rate for five. Nobody is handed one interest rate; what a depositor gets is a board, the board has a shape, and the shape is doing work: it says something about what the bank thinks longer money is worth to it. A term structure of interest rates is the same board, drawn for one borrower whose standing does not change as the horizon lengthens.

What each recorded SPOT rate actually does to a rupee

A rate is easier to hold on to once it is seen acting. Rs 1,000/- placed at each recorded SPOT rate, on annual compounding, comes back as one lump at the stated date. Nothing is paid along the way, and that is exactly what makes it a SPOT rate. The table below is the worked instance the rest of this guide returns to, and every figure in it is derived from the six recorded rates rather than looked up.

Two things in that table are worth pausing on. The first is that the thirty year figure is not thirty times, or six times, or any tidy multiple of the one year figure. Rs 1,000/- placed for one year at the one year SPOT rate returns Rs 1,059.000000/-, and six times that is Rs 6,354.000000/-. The thirty year SPOT rate returns Rs 9,002.603850/-. Compounding is doing far more of the work at the long end than the extra 1.70 percentage points of rate is. The second is that six decimal places are printed on purpose, so that a reader repeating the arithmetic lands on the same digits rather than wondering whether they made a mistake.

Where does the rate at the front of the term structure come from?

Not from the same place the rest of it comes from, and that asymmetry is the single most useful thing to carry out of this section. The shortest borrowing in any market is not priced the way a thirty year commitment is priced. At the very front sits an administered rateA rate whose level is announced by an authority rather than arrived at by two parties settling terms between themselves. The level, and the process behind it, belong to whoever publishes them., which is a rate somebody decides and announces rather than a rate that emerges when a borrower and a lender agree terms. Around it sits a band of further administered rates, and around that a set of operations that carry the decided level into the money marketThe part of the borrowing world where money changes hands for stretches measured in days and weeks rather than in years..

In India every part of that apparatus is set by the Reserve Bank of India at rbi.org.in. The level itself, the process by which it is decided, the band around it, what each rate in that band is used for, the operations that carry it outward, and the requirement attaching to a bank's deposits are all theirs to publish, and all of them move. Each of those levels is the Reserve Bank of India's to publish. Naming the item and routing it to the authority survives the next change; printing a level does not, and a reader who trusted a printed level would carry a stale figure into arithmetic that then looks perfectly sound and is not.

There is a second reason for leaving it empty, and it is arithmetical rather than editorial. An administered rate acts on borrowing measured in nights and weeks. The shortest node this record carries is one year. A decision by an authority therefore touches a market entirely inside the first year, off the left hand edge of everything drawn here. Sketching a plausible line into that space and reading a number off it would be inventing every point on it.

The everyday version. One notice board goes up at the entrance of a wholesale market setting the rate at which the market operator will lend for the night. The notice is decided and posted. A stall three rows in charges a customer for six months of credit, and that price is a different question with a different answer that nobody posted anywhere. The front of a term structure of interest rates is the notice; everything further out is what the stalls settled among themselves.

What survives the journey from the front of the curve to the ten year SPOT rate?

One quantity is what all four readings compete to explain, so it is worth computing before any of them is named. On the invented curve the ten year SPOT rate of 7.35 per cent a year stands above the one year SPOT rate of 5.90 per cent by 1.45 percentage points. Written in the other unit that is 145 basis pointsOne hundredth of one percentage point. A hundred of them make one percentage point, so 145 of them are 1.45 percentage points and not 145 of anything larger.. Percentage points and basis points are one quantity in two units, and an account that lets the two drift becomes wrong by a factor of a hundred without a single sum changing.

Now the important sentence about that 145 basis points. Nobody set it. It is not a decision, it is not published anywhere, and no authority signed it off. The 145 basis points is what remains after a rate decided at the very front has travelled outward along every length of time and been altered on the way by whatever the four readings are arguing about. A quantity starts in one place and shows up, changed, somewhere else, and the size of the change is 145 basis points. Four accounts of that journey follow.

The same distance measured to the far end is larger, and it is worth having both numbers. The thirty year SPOT rate of 7.60 per cent a year stands 1.70 percentage points above the one year SPOT rate of 5.90 per cent, or 170 basis points. The curve therefore rises at every single one of its five recorded steps, five out of five, with no reversal anywhere. The rise at every step matters later. One of the four readings has real trouble with it.

What does the expectations reading say that rising shape means?

The expectations reading says the simplest possible thing, and the simplicity is most of its appeal. On the expectations reading, the ten year SPOT rate is what it is because the market expects short rates to average roughly that much across the ten years. A rising term structure of interest rates therefore means one thing and one thing only: short rates are expected to be higher later than they are now. Nothing else is needed. No compensation, no preferences, no habits. Arithmetic and an expectation, and the shape falls out.

The household version follows the logic through exactly rather than merely illustrating it. A household has money to place for ten years. The household may place the money once at the ten year SPOT rate, or place it for a year, take it back, place it again, and repeat ten times. On the pure expectations reading those two routes must be expected to finish in the same place. If rolling were expected to finish ahead, everybody would roll and the one year rate would be bid until the advantage vanished. So the ten year SPOT rate has to be, in effect, the expected average of the ten one year rates that would be met along the way.

Now the limit, stated in the same block because a mechanism without its boundary teaches half a tool. On the pure form of this reading, a person who commits for ten years and a person who rolls ten times are expected to end up with exactly the same amount. Nobody is paid anything at all for accepting the longer commitment. That is not a small assumption dressed as a conclusion. The assumption is a strong claim about how people feel about being locked in, and the reading has no room anywhere in it for the possibility that they mind.

The reading is also the oldest and most diffuse of the four, and it does not sit with a single named author the way the other three do. The older account is the fairest name for it.

What does the liquidity preference reading add, and what does it give up to add it?

The liquidity preference reading keeps every bit of the expectations arithmetic and inserts one extra quantity. The claim is that a holder does mind being locked in, so the long SPOT rate carries an additional amount on top of the expected average of short rates, as compensation for tying money up over a longer stretch. The rising shape can then come from an expected rise, or from that compensation, or from any combination of the two. The extra amount has a name of its own, the term premium, and why it cannot be measured from a curve alone is settled separately.

The liquidity preference reading is attributed to John Hicks. A misattributed reading of the term structure of interest rates is exactly the kind of error a reader carries around for years without ever being corrected.

Here is the cost of the addition, and it is severe. Before the extra quantity was inserted, one observed number implied one expectation. After it, one observed number has to be split between two quantities that are both unobservable. The moment two hidden amounts sit inside one observed rate, the same observed rate can be produced by endlessly many pairs of them, and the curve stops being able to say which pair is at work. The reading became more plausible and less testable in one move, and both halves of that trade are real.

One equation holds two unknowns, and no amount of staring at the curve turns it into two. A reading that says most of the 1.45 percentage points is an expected rise with very little compensation fits the recorded 7.35 per cent exactly. So does a reading that says almost none of it is an expected rise and nearly all of it is compensation. Both were built to reproduce the observation, and both do. Splitting them needs something from outside the curve, and this record carries no expectation of any future rate anywhere in it, so the split cannot be made.

What does the market segmentation reading say instead, and where does this curve resist it?

The market segmentation reading drops the idea that the term structure of interest rates is one market at all. On the market segmentation reading, different holders work at different lengths of time for reasons that have nothing whatever to do with expectations about rates. An insurer with thirty year obligations lives at the far end because that is where its obligations are. A treasury desk parking cash lives at the near end because that is where its cash is needed. Neither is choosing a length of time by comparing rates, and neither will move.

If that is true, then the rate at each length of time is settled by what is wanted and what is available at that length of time alone, and neighbouring lengths of time need not connect to one another at all. The reading has one genuine strength that none of the others has: it explains a kink. If one particular length of time is crowded with holders and the one beside it is deserted, a segmented market produces a step that no smooth account of expectations can produce.

Holding that claim against the recorded curve calls for dividing rather than subtracting. A raw step between two recorded nodes hides how much time it covers. The step from the one year SPOT rate to the two year SPOT rate is 35 basis points and covers a single year. The step from the ten year SPOT rate to the thirty year SPOT rate is 25 basis points and covers twenty years. The two steps look comparable and are nothing of the kind.

Read the bars from left to right. Per year of extra waiting the recorded steps charge 0.3500 percentage points, then 0.3000, then 0.1750, then 0.0900, then 0.0125. Every one of the five is smaller than the one before it, five falls out of five possible, with no reversal at any point. A set of genuinely unconnected markets has no reason on earth to produce a sequence that falls smoothly at every single step, and this one does. The near end charges exactly twenty eight times what the far end charges for one more year of waiting, and the descent between them is orderly rather than jumbled.

So the reading is not refuted, but this particular curve does not support it. The reading needs a broken pattern to show itself, and the record here holds a graded one. The market segmentation reading is attributed to John Culbertson.

What does the preferred habitat reading do with the other three?

The preferred habitat reading takes the useful half of each and glues them together, and most people actually hold this one. The claim is that holders do have a length of time they would rather be at, exactly as the market segmentation reading says, but that they are not welded to it. Holders will move away from their preferred stretch if the rate somewhere else pays them enough to make the move worth the bother. Everybody has a habitat and everybody has a price for leaving it.

Notice what that buys. A large enough gap between two neighbouring lengths of time pulls somebody across and closes it, so the curve stays connected. The expectations reading needs exactly that to work at all. But a stretch crowded with willing holders and a deserted stretch beside it can stay a little out of line for as long as the gap is smaller than the cost of moving, so the curve can also carry lumps. Connected and lumpy at the same time. Most real schedules of rates look exactly like that.

The preferred habitat reading is attributed to Franco Modigliani and Richard Sutch.

Its limit is the same one that catches every reading in this guide. The preferred habitat reading predicts that some compensation exists without ever saying how much. No observed curve can contradict it. A prediction that survives every possible observation has said nothing about the world; it has said something about the shape of the sentence. The weakness is genuine, and being intuitively the most reasonable of the four does not repair it.

What does a parallel shift do to the term structure, and what does it deliberately leave alone?

A parallel shift is a movement in which every recorded SPOT rate changes by the same amount. Not roughly the same amount, not the same proportion, the same amount. A parallel shift is the simplest movement anybody defines on a term structure of interest rates, and its whole content is what it does not touch.

Work it on the invented curve, with the size declared as an input rather than observed anywhere. Add 50 basis points to every recorded node. The one year SPOT rate reads 6.40 per cent a year instead of 5.90. The two year SPOT rate reads 6.75 instead of 6.25. The three year reads 7.05, the five year reads 7.40, the ten year SPOT rate reads 7.85 instead of 7.35, and the thirty year reads 8.10. Six numbers, six identical additions, no exceptions.

Now measure the shape rather than the level, and this is the whole lesson. Before the shift, the ten year SPOT rate less the two year SPOT rate is 7.35 less 6.25, a distance of 1.10 percentage points or 110 basis points. After the shift the same subtraction is 7.85 less 6.75, a distance of 1.10 percentage points or 110 basis points. The level of the term structure of interest rates has moved and its shape has not moved at all, and that is not a happy accident but the entire definition of a parallel shift.

The cancellation is forced arithmetic rather than an interesting property of these particular numbers. The added amount appears twice in the subtraction with opposite signs, so it holds for any pair of nodes, any starting curve and any size of shift. Which is precisely why a parallel shift is the least informative movement a term structure of interest rates can make: it carries no news about shape whatsoever, and every one of the four readings in this guide is a claim about shape.

The everyday version is easy and, unusually, exact. Every price in one shop goes up by ten rupees on the same morning. The shop is dearer, and yet the order of what is dearest inside it is precisely what it was, and every gap between any two items is precisely what it was. Nothing at all has been learned about the shop by observing the ten rupees; what the observation gives is the level and nothing about the arrangement.

Why does a rating action leave every recorded SPOT rate exactly where it was?

Because it lands on a different object entirely. A rating action is a change in the published assessment that a rating agencyA business that publishes an assessment of how likely a named borrower is to pay what it promised. The meaning of its scale, and what it must publish when an assessment changes, are set by the regulator. makes about one named borrower. A rating action changes what that one borrower has to pay above a government rate of the same length of time. The government rates themselves do not change, and the term structure of interest rates here is made entirely of those.

Work the boundary with the invented issuer used across these subjects. Palash Cements Limited issues at five years at 9.10 per cent a year. The five year government securityA borrowing instrument issued by the government itself. Who may hold or deal in one, and how one first reaches a holder, are set by the authority rather than described here. SPOT rate on the invented curve is 6.90 per cent. The distance between them is 2.20 percentage points, or 220 basis points, and that distance is a credit spreadThe extra rate a borrower other than the government pays over a government rate of the same length of time. The reason for the compensation, and how a spread is read, are settled separately.. A credit spread and what it pays for are settled separately. Here the spread is only the object a rating action lands on.

Palash Cements Limited has no rating anywhere in this record. Whatever a rating action does, it does it to the 220 basis points and not to the 6.90 per cent underneath. The six recorded SPOT rates are the same six numbers the day before the action and the day after it.

Which of the four readings does this one curve settle?

None of them, and the reason is arithmetical rather than a failure of nerve. Set the four side by side against the same test, the 145 basis points standing between the one year SPOT rate and the ten year SPOT rate. The expectations reading accounts for the whole of it with short rates expected to average higher. The liquidity preference reading accounts for it with a smaller expected rise and compensation on top. The market segmentation reading accounts for it with what is wanted and available at each length of time separately. The preferred habitat reading accounts for it with some of each and a habit besides.

All four fit 145 basis points exactly, and they fit it exactly because each of them was built to. The curve reports one number for each length of time, and every reading except the pure expectations one needs at least two quantities inside that number. One observation, two or more unknowns, and no further observations anywhere in this record to bring to bear. The limit is not a shortage of effort but a shortage of observations.

The four are not four points on one line, so seeing where they sit relative to one another helps more than ranking them. The readings differ along two separate questions. The first is how connected they treat neighbouring lengths of time: fully joined at one extreme, entirely independent at the other. The second is how much of a long SPOT rate they treat as payment for the commitment rather than as expectation. Place the four against those two questions and the arrangement is informative.

One thing that map makes obvious is worth saying out loud. The expectations reading and the liquidity preference reading sit at the same place on the horizontal question and at opposite ends of the vertical one. They disagree about a quantity the curve never reports and agree about the one it does, and that is exactly why the curve cannot tell them apart. Two readings that differ only in an unobserved quantity are not two hypotheses a single observation can choose between.

Who actually reaches for a term structure of interest rates, and what do they take from it?

Three sorts of person, and each takes something different out of the same six numbers. The first is the one closest to home. A household with a lump sum and no immediate need for it faces the exact choice the expectations reading describes: place it once for a long stretch, or place it short and keep renewing. The board of deposit rates in front of them has a shape, and the shape is the only public information they have about what longer money is being priced at. The shape does not say which of the four readings produced it. So it does not say whether the longer rate is a forecast or a payment for being locked in, and no curve settles the choice between the two.

Second, somebody analysing an instrument that pays on several different dates. The whole reason to hold a term structure of interest rates rather than one rate is that a payment due in two years and a payment due in ten years are not the same length of commitment and should not be discounted at the same number. The two year SPOT rate of 6.25 per cent a year belongs to the first and the ten year SPOT rate of 7.35 per cent belongs to the second, and the arithmetic of putting those two together into one price is settled separately. The structure is the input to that work, and using one rate where six are available is where a great deal of quiet error enters.

Third, somebody with borrowing to arrange. A treasurer deciding whether to borrow for three years or for ten is looking at the same 145 basis points from the other side of the table, and that distance is a real cost of the longer arrangement. But the four readings apply from this side too. If the distance is mostly an expected rise, then borrowing short and refinancing is expected to cost the same in the end. If it is mostly payment for a longer commitment, borrowing short is expected to cost less and buys a different risk instead. The curve prices the choice and does not make it, and neither reading of the distance says which risk is the cheaper one to carry.

All three share a single discipline. The shape is read first and any single level second. A SPOT rate without its length of time attached is not a number anybody can use, so which of the six recorded lengths is meant has to be said every single time a rate is quoted. And when somebody says what the shape means, ask which of the four readings they are using. The odds are very good that they have chosen one without noticing that a choice was on offer.